For SonEd2, you have to merge the two files. Open a command prompt, navigate to the directory with the two files and type this:
Obviously replace those filenames with the actual ones.

The red screen of death is caused by a bad checksum, either fix the checksum (ESE has a tool to do this built in) or check the "ignore checksum" option in your emulator if it has one.

The flowers are sprites, so they won't appear correctly in the editor, I'm not sure about the loops though, it's been a long time since I hacked Sonic 1. To compile the level you need to build a new ROM using one of the disassemblies. I thought we had the disassembies in the database but I guess not, I'll add them later, for now you can get one here, you want ASM+BIN. Just split your ROM, insert your edited level in the correct folder, rename it (SonEd2 and this disassembly use different filenames), and run build.bat.

As for your loop issue in ESE2, only one of the loop tiles works properly, see which one the original game uses.

I've never used Sonik Sprite Remix so I can't help you with that, sorry.